Methods and apparatus for compressing and decompressing a higher order ambisonics representation

ABSTRACT

Higher Order Ambisonics represents three-dimensional sound independent of a specific loudspeaker set-up. However, transmission of an HOA representation results in a very high bit rate. Therefore, compression with a fixed number of channels is used, in which directional and ambient signal components are processed differently. The ambient HOA component is represented by a minimum number of HOA coefficient sequences. The remaining channels contain either directional signals or additional coefficient sequences of the ambient HOA component, depending on what will result in optimum perceptual quality. This processing can change on a frame-by-frame basis.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is division of U.S. patent application Ser. No.15/876,442, filed Jan. 22, 2018, which is division of Ser. No.15/650,674, filed Jul. 14, 2017, now U.S. Pat. No. 9,913,063, which iscontinuation of Ser. No. 14/787,978, filed Oct. 29, 2015, now U.S. Pat.No. 9,736,607, which is U.S. National Stage of International ApplicationNo. PCT/EP2014/058380, filed Apr. 24, 2014, which claims priority toEuropean Patent Application No. 13305558.2, filed Apr. 29, 2013, each ofwhich is incorporated by reference in its entirety.

TECHNICAL FIELD

The invention relates to a method and to an apparatus for compressingand decompressing a Higher Order Ambisonics representation by processingdirectional and ambient signal components differently.

BACKGROUND

Higher Order Ambisonics (HOA) offers one possibility to representthree-dimensional sound among other techniques like wave field synthesis(WFS) or channel based approaches like 22.2. In contrast to channelbased methods, however, the HOA representation offers the advantage ofbeing independent of a specific loudspeaker set-up. This flexibility,however, is at the expense of a decoding process which is required forthe playback of the HOA representation on a particular loudspeakerset-up. Compared to the WFS approach, where the number of requiredloudspeakers is usually very large, HOA may also be rendered to set-upsconsisting of only few loudspeakers. A further advantage of HOA is thatthe same representation can also be employed without any modificationfor binaural rendering to head-phones.

HOA is based on the representation of the spatial density of complexharmonic plane wave amplitudes by a truncated Spherical Harmonics (SH)expansion. Each expansion coefficient is a function of angularfrequency, which can be equivalently represented by a time domainfunction. Hence, without loss of generality, the complete HOA soundfield representation actually can be assumed to consist of O time domainfunctions, where O denotes the number of expansion coefficients. Thesetime domain functions will be equivalently referred to as HOAcoefficient sequences or as HOA channels.

The spatial resolution of the HOA representation improves with a growingmaximum order N of the expansion. Unfortunately, the number of expansioncoefficients O grows quadratically with the order N, in particularO=(V+1)². For example, typical HOA representations using order N=4require O=25 HOA (expansion) coefficients. According to the previouslymade considerations, the total bit rate for the transmission of HOArepresentation, given a desired single-channel sampling rate f_(s) andthe number of bits N_(b) per sample, is determined by O·f_(s)·N_(b).Consequently, transmitting an HOA representation of order N=4 with asampling rate of f_(s)=48 kHz employing N_(b)=16 bits per sample resultsin a bit rate of 19.2 MBits/s, which is very high for many practicalapplications, e.g. for streaming.

Compression of HOA sound field representations is proposed in patentapplications EP 12306569.0 and EP 12305537.8. Instead of perceptuallycoding each one of the HOA coefficient sequences individually, as it isperformed e.g. in E. Hellerud, I. Burnett, A. Solvang and U. P.Svensson, “Encoding Higher Order Ambisonics with AAC”, 124th AESConvention, Amsterdam, 2008, it is attempted to reduce the number ofsignals to be perceptually coded, in particular by performing a soundfield analysis and decomposing the given HOA representation into adirectional and a residual ambient component. The directional componentis in general supposed to be represented by a small number of dominantdirectional signals which can be regarded as general plane wavefunctions. The order of the residual ambient HOA component is reducedbecause it is assumed that, after the extraction of the dominantdirectional signals, the lower-order HOA coefficients are carrying themost relevant information.

SUMMARY OF INVENTION

Altogether, by such operation the initial number (N+1)² of HOAcoefficient sequences to be perceptually coded is reduced to a fixednumber of D dominant directional signals and a number of (N_(RED)+1)²HOA coefficient sequences representing the residual ambient HOAcomponent with a truncated order N_(RED)<N, whereby the number ofsignals to be coded is fixed, i.e. D+(N_(RED)+1)². In particular, thisnumber is independent of the actually detected number D_(ACT)(k)≤D ofactive dominant directional sound sources in a time frame k. This meansthat in time frames k, where the actually detected number D_(ACT)(k) ofactive dominant directional sound sources is smaller than the maximumallowed number D of directional signals, some or even all of thedominant directional signals to be perceptually coded are zero.Ultimately, this means that these channels are not used at all forcapturing the relevant information of the sound field.

In this context, a further possibly weak point in the EP 12306569.0 andEP 12305537.8 processings is the criterion for the determination of theamount of active dominant directional signals in each time frame,because it is not attempted to determine an optimal amount of activedominant directional signals with respect to the successive perceptualcoding of the sound field. For instance, in EP 12305537.8 the amount ofdominant sound sources is estimated using a simple power criterion,namely by determining the dimension of the subspace of theinter-coefficients correlation matrix belonging to the greatesteigenvalues. In EP 12306569.0 an incremental detection of dominantdirectional sound sources is proposed, where a directional sound sourceis considered to be dominant if the power of the plane wave functionfrom the respective direction is high enough with respect to the firstdirectional signal. Using power based criteria like in EP 12306569.0 andEP 12305537.8 may lead to a directional-ambient decomposition which issuboptimal with respect to perceptual coding of the sound field.

A problem to be solved by the invention is to improve HOA compression bydetermining for a current HOA audio signal content how to assign to apredetermined reduced number of channels, directional signals andcoefficients for the ambient HOA component.

The invention improves the compression processing proposed in EP12306569.0 in two aspects. First, the bandwidth provided by the givennumber of channels to be perceptually coded is better exploited. In timeframes where no dominant sound source signals are detected, the channelsoriginally reserved for the dominant directional signals are used forcapturing additional information about the ambient component, in theform of additional HOA coefficient sequences of the residual ambient HOAcomponent. Second, having in mind the goal to exploit a given number ofchannels to perceptually code a given HOA sound field representation,the criterion for the determination of the amount of directional signalsto be extracted from the HOA representation is adapted with respect tothat purpose. The number of directional signals is determined such thatthe decoded and reconstructed HOA representation provides the lowestperceptible error. That criterion compares the modelling errors arisingeither from extracting a directional signal and using a HOA coefficientsequence less for describing the residual ambient HOA component, orarising from not extracting a directional signal and instead using anadditional HOA coefficient sequence for describing the residual ambientHOA component. That criterion further considers for both cases thespatial power distribution of the quantisation noise introduced by theperceptual coding of the directional signals and the HOA coefficientsequences of the residual ambient HOA component.

In order to implement the above-described processing, before startingthe HOA compression, a total number 1 of signals (channels) is specifiedcompared to which the original number of O HOA coefficient sequences isreduced. The ambient HOA component is assumed to be represented by aminimum number O_(RED) of HOA coefficient sequences. In some cases, thatminimum number can be zero. The remaining D=I−O_(RED) channels aresupposed to contain either directional signals or additional coefficientsequences of the ambient HOA component, depending on what thedirectional signal extraction processing decides to be perceptually moremeaningful. It is assumed that the assigning of either directionalsignals or ambient HOA component coefficient sequences to the remainingD channels can change on frame-by-frame basis. For reconstruction of thesound field at receiver side, information about the assignment istransmitted as extra side information.

In principle, the inventive compression method is suited for compressingusing a fixed number of perceptual encodings a Higher Order Ambisonicsrepresentation of a sound field, denoted HOA, with input time frames ofHOA coefficient sequences, said method including the following stepswhich are carried out on a frame-by-frame basis:

for a current frame, estimating a set of dominant directions and acorresponding data set of indices of detected directional signals;

decomposing the HOA coefficient sequences of said current frame into anon-fixed number of directional signals with respective directionscontained in said set of dominant direction estimates and with arespective data set of indices of said directional signals, wherein saidnon-fixed number is smaller than said fixed number,

and into a residual ambient HOA component that is represented by areduced number of HOA coefficient sequences and a corresponding data setof indices of said reduced number of residual ambient HOA coefficientsequences, which reduced number corresponds to the difference betweensaid fixed number and said non-fixed number;

assigning said directional signals and the HOA coefficient sequences ofsaid residual ambient HOA component to channels the number of whichcorresponds to said fixed number, wherein for said assigning said dataset of indices of said directional signals and said data set of indicesof said reduced number of residual ambient HOA coefficient sequences areused;

perceptually encoding said channels of the related frame so as toprovide an encoded compressed frame.

In principle the inventive compression apparatus is suited forcompressing using a fixed number of perceptual encodings a Higher OrderAmbisonics representation of a sound field, denoted HOA, with input timeframes of HOA coefficient sequences, said apparatus carrying out aframe-by-frame based processing and including:

means being adapted for estimating for a current frame a set of dominantdirections and a corresponding data set of indices of detecteddirectional signals;

means being adapted for decomposing the HOA coefficient sequences ofsaid current frame into a non-fixed number of directional signals withrespective directions contained in said set of dominant directionestimates and with a respective data set of indices of said directionalsignals, wherein said non-fixed number is smaller than said fixednumber,

and into a residual ambient HOA component that is represented by areduced number of HOA coefficient sequences and a corresponding data setof indices of said reduced number of residual ambient HOA coefficientsequences, which reduced number corresponds to the difference betweensaid fixed number and said non-fixed number;

means being adapted for assigning said directional signals and the HOAcoefficient sequences of said residual ambient HOA component to channelsthe number of which corresponds to said fixed number, wherein for saidassigning said data set of indices of said directional signals and saiddata set of indices of said reduced number of residual ambient HOAcoefficient sequences are used;

means being adapted for perceptually encoding said channels of therelated frame so as to provide an encoded compressed frame.

In principle, the inventive decompression method is suited fordecompressing a Higher Order Ambisonics representation compressedaccording to the above compression method, said decompressing includingthe steps:

perceptually decoding a current encoded compressed frame so as toprovide a perceptually decoded frame of channels;

re-distributing said perceptually decoded frame of channels, using saiddata set of indices of detected directional signals and said data set ofindices of the chosen ambient HOA coefficient sequences, so as torecreate the corresponding frame of directional signals and thecorresponding frame of the residual ambient HOA component;

re-composing a current decompressed frame of the HOA representation fromsaid frame of directional signals and from said frame of the residualambient HOA component, using said data set of indices of detecteddirectional signals and said set of dominant direction estimates,wherein directional signals with respect to uniformly distributeddirections are predicted from said directional signals, and thereaftersaid current decompressed frame is re-composed from said frame ofdirectional signals, said predicted signals and said residual ambientHOA component.

In principle the inventive decompression apparatus is suited fordecompressing a Higher Order Ambisonics representation compressedaccording to the above compression method, said apparatus including:

means being adapted for perceptually decoding a current encodedcompressed frame so as to provide a perceptually decoded frame ofchannels;

means being adapted for re-distributing said perceptually decoded frameof channels, using said data set of indices of detected directionalsignals and said data set of indices of the chosen ambient HOAcoefficient sequences, so as to recreate the corresponding frame ofdirectional signals and the corresponding frame of the residual ambientHOA component;

means being adapted for re-composing a current decompressed frame of theHOA representation from said frame of directional signals, said frame ofthe residual ambient HOA component, said data set of indices of detecteddirectional signals, and said set of dominant direction estimates,

wherein directional signals with respect to uniformly distributeddirections are predicted from said directional signals, and thereaftersaid current decompressed frame is re-composed from said frame ofdirectional signals, said predicted signals and said residual ambientHOA component.

In one example, a method for decompressing a compressed Higher OrderAmbisonics representation, includes

perceptually decoding a current encoded compressed frame to provide aperceptually decoded frame of channels;

re-distributing said perceptually decoded frame of channels based on anassignment vector indicating at least an index of a possibly containedcoefficient sequence of an ambient HOA component and a data set ofindices of directional signals in order to determine a correspondingframe of the ambient HOA component;

re-composing a current decompressed frame of the HOA representation fromthe recreated frame of directional signals and from the recreated frameof the ambient HOA component based on a data set of indices of detecteddirectional signals and a set of dominant direction estimates,

wherein directional signals with respect to uniformly distributeddirections are predicted from said directional signals, and thereaftersaid current decompressed frame is re-composed from the recreated frameof directional signals, said predicted signals and said ambient HOAcomponent.

In one example, an apparatus for decompressing a Higher Order Ambisonicsrepresentation compressed, said apparatus including:

means adapted for perceptually decoding a current encoded compressedframe so as to provide a perceptually decoded frame of channels;

means adapted for re-distributing said perceptually decoded frame ofchannels based on an assignment vector indicating at least an index of apossibly contained coefficient sequence of an ambient HOA component anda data set of indices of directional signals in order to determine acorresponding frame of the ambient HOA component;

means adapted for re-composing a current decompressed frame of the HOArepresentation from the recreated frame of directional signals and fromthe recreated frame of the ambient HOA component based on a data set ofindices of detected directional signals and a set of dominant directionestimates,

wherein directional signals with respect to uniformly distributeddirections are predicted from said directional signals, and thereaftersaid current decompressed frame is re-composed from the recreated frameof directional signals, said predicted signals and said ambient HOAcomponent.

BRIEF DESCRIPTION OF DRAWINGS

Exemplary embodiments of the invention are described with reference tothe accompanying drawings, which show in:

FIG. 1 illustrates block diagram for the HOA compression;

FIG. 2 illustrates estimation of dominant sound source directions;

FIG. 3 illustrates block diagram for the HOA decompression;

FIG. 4 illustrates spherical coordinate system;

FIG. 5 illustrates normalised dispersion function v_(N)(θ) for differentAmbisonics orders N and for angles θ∈[0,π].

DESCRIPTION OF EMBODIMENTS A. Improved HOA Compression

The compression processing according to the invention, which is based onEP 12306569.0, is illustrated in FIG. 1 where the signal processingblocks that have been modified or newly introduced compared to EP12306569.0 are presented with a bold box, and where ‘

’ (direction estimates as such) and ‘C’ in this application correspondto ‘A’ (matrix of direction estimates) and ‘D’ in EP 12306569.0,respectively.

For the HOA compression a frame-wise processing with non-overlappinginput frames C(k) of HOA coefficient sequences of length L is used,where k denotes the frame index. The frames are defined with respect tothe HOA coefficient sequences specified in equation (45) as

C(k): =[c((kL+1)T _(S)) c((kL+2)T _(S)) c((k+1)LT _(S))],  (1)

where T_(S) indicates the sampling period.The first step or stage 11/12 in FIG. 1 is optional and consists ofconcatenating the non-overlapping k-th and the (k−1)-th frames of HOAcoefficient sequences into a long frame t(k) as

{tilde over (C)}(k):=[C(k−1)C(k)],  (2)

which long frame is 50% overlapped with an adjacent long frame and whichlong frame is successively used for the estimation of dominant soundsource directions. Similar to the notation for {tilde over (C)}(k), thetilde symbol is used in the following description for indicating thatthe respective quantity refers to long overlapping frames. If step/stage11/12 is not present, the tilde symbol has no specific meaning.

In principle, the estimation step or stage 13 of dominant sound sourcesis carried out as proposed in EP 13305156.5, but with an importantmodification. The modification is related to the determination of theamount of directions to be detected, i.e. how many directional signalsare supposed to be extracted from the HOA representation. This isaccomplished with the motivation to extract directional signals only ifit is perceptually more relevant than using instead additional HOAcoefficient sequences for better approximation of the ambient HOAcomponent. A detailed description of this technique is given in sectionA.2.

The estimation provides a data set

_(DIR,ACT)(k)⊆{1, . . . , D} of indices of directional signals that havebeen detected as well as the set

_(Ω,ACT)(k) of corresponding direction estimates. D denotes the maximumnumber of directional signals that has to be set before starting the HOAcompression.

In step or stage 14, the current (long) frame {tilde over (C)}(k) of HOAcoefficient sequences is decomposed (as proposed in EP 13305156.5) intoa number of directional signals X_(DIR)(k−2) belonging to the directionscontained in the set

_(Ω,ACT)(k), and a residual ambient HOA component C_(AMB)(k−2). Thedelay of two frames is introduced as a result of overlap-add processingin order to obtain smooth signals. It is assumed that X_(DIR)(k−2) iscontaining a total of D channels, of which however only thosecorresponding to the active directional signals are non-zero. Theindices specifying these channels are assumed to be output in the dataset

_(DIR,ACT)(k−2) Additionally, the decomposition in step/stage 14provides some parameters ζ(k−2) which are used at decompression side forpredicting portions of the original HOA representation from thedirectional signals (see EP 13305156.5 for more details).

In step or stage 15, the number of coefficients of the ambient HOAcomponent C_(AMB)(k−2) is intelligently reduced to contain onlyO_(RED)+D−N_(DIR,ACT)(k−2) non-zero HOA coefficient sequences, whereN_(DIR,ACT)(k−2)=|

_(DIR,ACT)(k−2)| indicates the cardinality of the data set

_(DIR,ACT)(k−2), i.e. the number of active directional signals in framek−2. Since the ambient HOA component is assumed to be always representedby a minimum number O_(RED) of HOA coefficient sequences, this problemcan be actually reduced to the selection of the remainingD−N_(DIR,ACT)(k−2) HOA coefficient sequences out of the possibleO−O_(RED) ones. In order to obtain a smooth reduced ambient HOArepresentation, this choice is accomplished such that, compared to thechoice taken at the previous frame k−3, as few changes as possible willoccur.

In particular, the three following cases are to be differentiated:

-   a) N_(DIR,ACT)(k−2)=N_(DIR,ACT)(k−3): In this case the same HOA    coefficient sequences are assumed to be selected as in frame k−3.-   b) N_(DIR,ACT)(k−2)<N_(DIR,ACT)(k−3): In this case, more HOA    coefficient sequences than in the last frame k−3 can be used for    representing the ambient HOA component in the current frame. Those    HOA coefficient sequences that were selected in k−3 are assumed to    be also selected in the current frame. The additional HOA    coefficient sequences can be selected according to different    criteria. For instance, selecting those HOA coefficient sequences in    C_(AMB)(k−2) with the highest average power, or selecting the HOA    coefficients sequences with respect to their perceptual    significance.-   c) N_(DIR,ACT)(k−2)>N_(DIR,ACT)(k−3): In this case, less HOA    coefficient sequences than in the last frame k−3 can be used for    representing the ambient HOA component in the current frame. The    question to be answered here is which of the previously selected HOA    coefficient sequences have to be deactivated. A reasonable solution    is to deactivate those sequences which were assigned to the channels    i∈    _(DIR,ACT)(k−2) at the signal assigning step or stage 16 at frame    k−3.

For avoiding discontinuities at frame borders when additional HOAcoefficient sequences are activated or deactivated, it is advantageousto smoothly fade in or out the respective signals.

The final ambient HOA representation with the reduced number ofO_(RED)+N_(DIR,ACT)(k−2) non-zero coefficient sequences is denoted byC_(AMB,RED)(k−2). The indices of the chosen ambient HOA coefficientsequences are output in the data set

_(AMB,ACT)(k−2).

In step/stage 16, the active directional signals contained inX_(DIR)(k−2) and the HOA coefficient sequences contained inC_(AMB,RED)(k−2) are assigned to the frame Y(k−2) of I channels forindividual perceptual encoding. To describe the signal assignment inmore detail, the frames X_(DIR)(k−2), Y(k−2) and C_(AMB,RED)(k−2) areassumed to consist of the individual signals x_(DIR,d)(k−2), d∈{1, . . ., D}, y_(i)(k−2), i∈{1, . . . , I} and c_(AMB,RED,o)(k−2), o∈{1, . . . ,0} as follows:

$\begin{matrix}{{{X_{DIR}\left( {k - 2} \right)} = \begin{bmatrix}{x_{{DIR},1}\left( {k - 2} \right)} \\{x_{{DIR},2}\left( {k - 2} \right)} \\\vdots \\{x_{{DIR},D}\left( {k - 2} \right)}\end{bmatrix}},{{C_{{AMB},{RED}}\left( {k - 2} \right)} = \begin{bmatrix}{c_{{AMB},{RED},1}\left( {k - 2} \right)} \\{c_{{AMB},{RED},2}\left( {k - 2} \right)} \\\vdots \\{c_{{AMB},{RED},O}\left( {k - 2} \right)}\end{bmatrix}},{{Y\left( {k - 2} \right)} = {\begin{bmatrix}{y_{1}\left( {k - 2} \right)} \\{y_{2}\left( {k - 2} \right)} \\\vdots \\{y_{I}\left( {k - 2} \right)}\end{bmatrix}.}}} & (3)\end{matrix}$

The active directional signals are assigned such that they keep theirchannel indices in order to obtain continuous signals for the successiveperceptual coding. This can be expressed by

y _(d)(k−2)=x _(DIR,d)(k−2) for all d∈

_(DIR,ACT)(k−2).  (4)

The HOA coefficient sequences of the ambient component are assigned suchthe minimum number of O_(RED) coefficient sequences is always containedin the last O_(RED) signals of Y(k−2), i.e.

y _(D+o)(k−2)=c _(AMB,RED,o)(k−2) for 1≤o≤O _(RED).  (5)

For the additional D−N_(DIR,ACT)(k−2) HOA coefficient sequences of theambient component it is to be differentiated whether or not they werealso selected in the previous frame:

-   a) If they were also selected to be transmitted in the previous    frame, i.e. if the respective indices are also contained in data set    _(AMB,ACT)(k−3), the assignment of these coefficient sequences to    the signals in Y(k−2) is the same as for the previous frame. This    operation assures smooth signals y_(i)(k−2), which is favourable for    the successive perceptual coding in step or stage 17.-   b) Otherwise, if some coefficient sequences are newly selected, i.e.    if their indices are contained in data set    _(AMB,ACT)(k−2) but not in data set    _(AMB,ACT)(k−3), they are first arranged with respect to their    indices in an ascending order and are in this order assigned to    channels i∉    _(DIR,ACT)(k−2) of Y(k−2) which are not yet occupied by directional    signals.    -   This specific assignment offers the advantage that, during a HOA        decompression process, the signal redistribution and composition        can be performed without the knowledge about which ambient HOA        coefficient sequence is contained in which channel of Y(k−2).        Instead, the assignment can be reconstructed during HOA        decompression with the mere knowledge of the data sets        _(AMB,ACT)(k−2) and        _(DIR,ACT)(k).

Advantageously, this assigning operation also provides the assignmentvector γ(k)∈

^(D-N) ^(DIR,ACT) ^((k−2)), whose elements γ_(o)(k), o=1, . . . ,D−N_(DIR,ACT)(k−2), denote the indices of each one of the additionalD−N_(DIR,ACT)(k−2) HOA coefficient sequences of the ambient component.To say it differently, the elements of the assignment vector γ(k)provide information about which of the additional O−O_(RED) HOAcoefficient sequences of the ambient HOA component are assigned into theD−N_(DIR,ACT)(k−2) channels with inactive directional signals. Thisvector can be transmitted additionally, but less frequently than by theframe rate, in order to allow for an initialisation of there-distribution procedure performed for the HOA decompression (seesection B). Perceptual coding step/stage 17 encodes the I channels offrame Y(k−2) and outputs an encoded frame Y̆(k−2).

For frames for which vector γ(k) is not transmitted from step/stage 16,at decompression side the data parameter sets

_(DIR,ACT)(k) and

_(AMB,ACT)(k−2) instead of vector γ(k) are used for the performing there-distribution.

A.1 Estimation of the Dominant Sound Source Directions

The estimation step/stage 13 for dominant sound source directions ofFIG. 1 is depicted in FIG. 2 in more detail. It is essentially performedaccording to that of EP 13305156.5, but with a decisive difference,which is the way of determining the amount of dominant sound sources,corresponding to the number of directional signals to be extracted fromthe given HOA representation. This number is significant because it isused for controlling whether the given HOA representation is betterrepresented either by using more directional signals or instead by usingmore HOA coefficient sequences to better model the ambient HOAcomponent.

The dominant sound source directions estimation starts in step or stage21 with a preliminary search for the dominant sound source directions,using the long frame {tilde over (C)}(k) of input HOA coefficientsequences. Along with the preliminary direction estimates {tilde over(Ω)}_(DOM) ^((d))(k), 1≤d≤D, the corresponding directional signals{tilde over (x)}_(DOM) ^((d))(k) and the HOA sound field components{tilde over (C)}_(DOM,CORR) ^((d))(k), which are supposed to be createdby the individual sound sources, are computed as described in EP13305156.5. In step or stage 22, these quantities are used together withthe frame {tilde over (C)}(k) of input HOA coefficient sequences fordetermining the number {tilde over (D)}(k) of directional signals to beextracted. Consequently, the direction estimates {tilde over (Ω)}_(DOM)^((d))(k), {tilde over (D)}(k)<d≤D, the corresponding directionalsignals {tilde over (x)}_(DOM) ^((d))(k), and HOA sound field components{tilde over (C)}_(DOM,CORR) ^((d))(k) are discarded. Instead, only thedirection estimates {tilde over (Ω)}_(DOM) ^((d))(k), 1≤d≤{tilde over(D)}(k) are then assigned to previously found sound sources.

In step or stage 23, the resulting direction trajectories are smoothedaccording to a sound source movement model and it is determined whichones of the sound sources are supposed to be active (see EP 13305156.5).The last operation provides the set

_(DIR,ACT)(k) of indices of active directional sound sources and the set

_(Ω,ACT)(k) of the corresponding direction estimates.

A.2 Determination of Number of Extracted Directional Signals

For determining the number of directional signals in step/stage 22, thesituation is assumed that there is a given total amount of I channelswhich are to be exploited for capturing the perceptually most relevantsound field information. Therefore, the number of directional signals tobe extracted is determined, motivated by the question whether for theoverall HOA compression/decompression quality the current HOArepresentation is represented better by using either more directionalsignals, or more HOA coefficient sequences for a better modelling of theambient HOA component.

To derive in step/stage 22 a criterion for the determination of thenumber of directional sound sources to be extracted, which criterion isrelated to the human perception, it is taken into consideration that HOAcompression is achieved in particular by the following two operations:

-   -   reduction of HOA coefficient sequences for representing the        ambient HOA component (which means reduction of the number of        related channels);    -   perceptual encoding of the directional signals and of the HOA        coefficient sequences for representing the ambient HOA        component.

Depending on the number M, 0≤M≤D, of extracted directional signals, thefirst operation results in the approximation

{tilde over (C)}(k)≈{tilde over (C)} ^((M))(k)  (6)

:={tilde over (C)} _(DIR) ^((M))(k)+{tilde over (C)} _(AMB,RED)^((M))(k),  (7)

where {tilde over (C)} _(DIR) ^((M))(k):=Σ_(d=1) ^(M) {tilde over (C)}_(DOM,CORR) ^((d))(k)  (8)

denotes the HOA representation of the directional component consistingof the HOA sound field components {tilde over (C)}_(DOM,CORR) ^((d))(k),1≤d≤M, supposed to be created by the M individually considered soundsources, and {tilde over (C)}_(AMB,RED) ^((M))(k) denotes the HOArepresentation of the ambient component with only I−M non-zero HOAcoefficient sequences.

The approximation from the second operation can be expressed by

{tilde over (C)}(k)≈{circumflex over ({tilde over (C)})}^((M))(k)  (9)

:={circumflex over ({tilde over (C)})}_(DIR) ^((M))(k)+{circumflex over({tilde over (C)})}_(AVB,RED) ^((M))(k)  (10)

where {circumflex over ({tilde over (C)})}_(DIR) ^((M))(k) and{circumflex over ({tilde over (C)})}_(AMB,RED) ^((M))(k) denote thecomposed directional and ambient HOA components after perceptualdecoding, respectively.

Formulation of Criterion

The number {tilde over (D)}(k) of directional signals to be extracted ischosen such that the total approximation error

{circumflex over ({tilde over (E)})}^((M))(k):={tilde over(C)}(k)−{circumflex over ({tilde over (C)})}^((M))(k)  (11)

with M={tilde over (D)}(k) is as less significant as possible withrespect to the human perception. To assure this, the directional powerdistribution of the total error for individual Bark scale critical bandsis considered at a predefined number Q of test directions Ω_(q), q=1, .. . , Q, which are nearly uniformly distributed on the unit sphere. Tobe more specific, the directional power distribution for the b-thcritical band, b=1, . . . , B, is represented by the vector

^((M))(k,b):=[

₁ ^((M))(k,b)

₂ ^((M))(k,b) . . .

_(Q) ^((M))(k,b)]^(T),  (12)

whose components

_(q) ^((M))(k,b) denote the power of the total error {circumflex over({tilde over (E)})}^((M))(k) related to the direction Ω_(q), the b-thBark scale critical band and the k-th frame. The directional powerdistribution

₁ ^((M))(k,b) of the total error {circumflex over ({tilde over(E)})}^((M))(k) is compared with the directional perceptual maskingpower distribution

{tilde over (P)} _(MASK)(k,b):=[{tilde over (P)} _(MASK,1)(k,b) {tildeover (P)} _(MASK,2)(k,b) . . . {tilde over (P)}_(MASK,Q)(k,b)]^(T)  (13)

due to the original HOA representation {tilde over (C)}(k). Next, foreach test direction Ω_(q) and critical band b the level of perception

_(q) ^((M))(k,b) of the total error is computed. It is here essentiallydefined as the ratio of the directional power of the total error{circumflex over ({tilde over (E)})}^((M))(k) and the directionalmasking power according to

$\begin{matrix}{{{{\overset{\sim}{\mathcal{L}}}_{q}^{(M)}\left( {k,b} \right)}\text{:}}\; = {{\max \left( {0,{\frac{{\hat{\overset{\sim}{}}}_{q}^{(M)}\left( {k,b} \right)}{{\overset{\sim}{}}_{{MASK},q}\left( {k,b} \right)} - 1}} \right)}.}} & (14)\end{matrix}$

The subtraction of ‘1’ and the successive maximum operation is performedto ensure that the perception level is zero, as long as the error poweris below the masking threshold.Finally, the number {tilde over (D)}(k) of directionals signals to beextracted can be chosen to minimise the average over all test directionsof the maximum of the error perception level over all critical bands,i.e.,

$\begin{matrix}{{\overset{\sim}{D}(k)} = {\underset{M}{argmin}\frac{1}{Q}{\sum\limits_{q = 1}^{Q}\; {\max\limits_{b}{{{\overset{\sim}{\mathcal{L}}}_{q}^{(M)}\left( {k,b} \right)}.}}}}} & (15)\end{matrix}$

It is noted that, alternatively, it is possible to replace the maximumby an averaging operation in equation (15).

Computation of the Directional Perceptual Masking Power Distribution

For the computation of the directional perceptual masking powerdistribution

_(MASK)(k,b) due to the original HOA representation {tilde over (C)}(k),the latter is transformed to the spatial domain in order to berepresented by general plane waves {tilde over (v)}_(q)(k) impingingfrom the test directions Ω_(q), q=1, . . . , Q. When arranging thegeneral plane wave signals {tilde over (v)}_(q)(k) in the matrix {tildeover (V)}(k) as

$\begin{matrix}{{{\overset{\sim}{V}(k)} = \begin{bmatrix}{{\overset{\sim}{v}}_{1}(k)} \\{{\overset{\sim}{v}}_{2}(k)} \\\vdots \\{{\overset{\sim}{v}}_{Q}(k)}\end{bmatrix}},} & (16)\end{matrix}$

the transformation to the spatial domain is expressed by the operation

{tilde over (V)}(k)=Ξ^(T) {tilde over (C)}(k),  (17)

where Ξ denotes the mode matrix with respect to the test directionΩ_(q), q=1, . . . , Q, defined by

Ξ:=[S ₁ S ₂ . . . S _(Q)]∈

^(O×Q)  (18)

with

[S ₀ ⁰(Ω_(q)) S ⁻¹ ⁻¹(Ω_(q)) S ⁻¹ ⁰(Ω_(q)) S ⁻ ¹(Ω_(q)) S ⁻² ⁻²(Ω_(q)) .. . S _(N) ^(N)(Ω_(q))]^(T)∈

^(O).  (19)

The elements

_(MASK)(k,b) of the directional perceptual masking power distribution

_(MASK)(k,b), due to the original HOA representation {tilde over(C)}(k), are corresponding to the masking powers of the general planewave functions {tilde over (v)}_(q)(k) for individual critical bands b.

Computation of Directional Power Distribution

In the following two alternatives for the computation of the directionalpower distribution

^((M))(k,b) are presented:

-   a. One possibility is to actually compute the approximation    {circumflex over ({tilde over (C)})}^((M))(k) of the desired HOA    representation {tilde over (C)}(k) by performing the two operations    mentioned at the beginning of section A.2. Then the total    approximation error {circumflex over ({tilde over (E)})}^((M))(k) is    computed according to equation (11). Next, the total approximation    error {circumflex over ({tilde over (E)})}^((M))(k) is transformed    to the spatial domain in order to be represented by general plane    waves {circumflex over ({tilde over (w)})}_(q) ^((M))(k) impinging    from the test directions Ω_(q), q=1, . . . , Q. Arranging the    general plane wave signals in the matrix {circumflex over ({tilde    over (W)})}^((M))(k) as

$\begin{matrix}{{{{\hat{\overset{\sim}{W}}}^{(M)}(k)} = \begin{bmatrix}{{\hat{\overset{\sim}{w}}}_{1}^{(M)}(k)} \\{{\hat{\overset{\sim}{w}}}_{2}^{(M)}(k)} \\\vdots \\{{\hat{\overset{\sim}{w}}}_{Q}^{(M)}(k)}\end{bmatrix}},} & (20)\end{matrix}$

the transformation to the spatial domain is expressed by the operation

{circumflex over ({tilde over (W)})}^((M))(k)=Ξ^(T){circumflex over({tilde over (E)})}^((M))(k).  (21)

The elements

_(q) ^((M))(k,b) of the directional power distribution

^((M))(k,b) of the total approximation error {circumflex over ({tildeover (E)})}^((M))(k) are obtained by computing the powers of the generalplane wave functions {circumflex over ({tilde over (w)})}^((M))(k), q=1,. . . , Q, within individual critical bands b.

-   b. The alternative solution is to compute only the approximation    {tilde over (C)}^((M))(k) instead of {circumflex over ({tilde over    (C)})}^((M))(k). This method offers the advantage that the    complicated perceptual coding of the individual signals needs not be    carried out directly. Instead, it is sufficient to know the powers    of the perceptual quantisation error within individual Bark scale    critical bands. For this purpose, the total approximation error    defined in equation (11) can be written as a sum of the three    following approximation errors:

{tilde over (E)} ^((M))(k):={tilde over (C)}(k)−{tilde over (C)}^((M))(k)  (22)

{circumflex over ({tilde over (E)})}_(DIR) ^((M))(k):={tilde over (C)}_(DIR) ^((M))(k)−{circumflex over ({tilde over (C)})}_(DIR(M))(k)  (23)

{circumflex over ({tilde over (E)})}_(AVB,RED) ^((M))(k):={tilde over(C)} _(AVB,RED) ^((M))(k)−{circumflex over ({tilde over (C)})}_(AVB,RED)^((M))(k),  (24)

which can be assumed to be independent of each other. Due to thisindependence, the directional power distribution of the total error{circumflex over ({tilde over (E)})}^((M))(k) can be expressed as thesum of the directional power distributions of the three individualerrors {tilde over (E)}^((M))(k), {circumflex over ({tilde over(E)})}_(DIR) ^((M))(k) and {circumflex over ({tilde over(E)})}_(AMB,RED) ^((M))(k).

The following describes how to compute the directional powerdistributions of the three errors for individual Bark scale criticalbands:

-   a. To compute the directional power distribution of the error {tilde    over (E)}^((M))(k), it is first transformed to the spatial domain by

{tilde over (W)} ^((M))(k)=↓^(T) {tilde over (E)} ^((M))(k),  (25)

wherein the approximation error {tilde over (E)}^((M))(k) is hencerepresented by general plane waves {tilde over (w)}_(q) ^((M))(k)impinging from the test directions Ω_(q), q=1, . . . , Q, which arearranged in the matrix {tilde over (W)}^((M))(k) according to

$\begin{matrix}{{{\overset{\sim}{W}}^{(M)}(k)} = {\begin{bmatrix}{{\overset{\sim}{w}}_{1}^{(M)}(k)} \\{{\overset{\sim}{w}}_{2}^{(M)}(k)} \\\vdots \\{{\overset{\sim}{w}}_{Q}^{(M)}(k)}\end{bmatrix}.}} & (26)\end{matrix}$

-   -   Consequently, the elements        _(q) ^((M))(k,b) of the directional power distribution        ^((M))(k,b) of the approximation error {tilde over (E)}^((M))(k)        are obtained by computing the powers of the general plane wave        functions {tilde over (w)}_(q) ^((M))(k), q=1, . . . , Q, within        individual critical bands b.

-   b. For computing the directional power distribution    _(DIR) ^((M))(k,b) of the error {circumflex over ({tilde over    (E)})}_(DIR) ^((M))(k), it is to be borne in mind that this error is    introduced into the directional HOA component {tilde over (C)}_(DIR)    ^((M))(k) by perceptually coding the directional signals {tilde over    (x)}_(DOM) ^((d))(k), 1≤d≤M. Further, it is to be considered that    the directional HOA component is given by equation (8). Then for    simplicity it is assumed that the HOA component {tilde over    (C)}_(DOM,CORR) ^((d))(k) is equivalently represented in the spatial    domain by O general plane wave functions {tilde over (v)}_(GRID,o)    ^((d))(k), which are created from the directional signal {tilde over    (x)}_(DOM) ^((d))(k) by a mere scaling, i.e.

{tilde over (v)} _(GRID,o) ^((d))(k)=α_(o) ^((d))(k){tilde over (x)}_(DOM) ^((d))(k),   (27)

where α_(o) ^((d))(k), o=1, . . . , O, denote the scaling parameters.The respective plane wave directions {tilde over (Ω)}_(ROT,o) ^((d))(k),0=1, . . . , O, are assumed to be uniformly distributed on the unitsphere and rotated such that {tilde over (Ω)}_(ROT,1) ^((d))(k)corresponds to the direction estimate {tilde over (Ω)}_(DOM) ^((d))(k).Hence, the scaling parameter α_(1(d))(k) is equal to ‘1’.

When defining Ξ_(GRID) ^((d))(k) to be the mode matrix with respect tothe rotated directions {tilde over (Ω)}_(ROT,o) ^((d))(k), o=1, . . . ,O, and arranging all scaling parameters α_(o) ^((d))(k) in a vectoraccording to

α^((d))(k):=[1 α₂ ^((d))(k) α₃ ^((d))(k) . . . α₀ ^((d))(k)]^(T)∈

^(O),  (28)

the HOA component {tilde over (C)}_(DOM,CORR) ^((d))(k) can be writtenas

{tilde over (C)} _(DOM,CORR) ^((d))(k)=Ξ_(GRID)^((d))(k)α^((d))(k){tilde over (x)} _(DOM) ^((d))(k).  (29)

Consequently, the error {circumflex over ({tilde over (E)})}_(DIR)^((M))(k) (see equation (23)) between the true directional HOA component

{tilde over (C)} _(DIR) ^((M))(k)=Σ_(d=1) ^(M) {tilde over (C)}_(DOM,CORR) ^((d))(k)  (30)

and that composed from the perceptually decoded directional signals{circumflex over ({tilde over (x)})}_(DOM) ^((d))(k), d=1, . . . , M, by

$\begin{matrix}{{{\hat{\overset{\sim}{C}}}_{DIR}^{(M)}(k)} = {\sum\limits_{d = 1}^{M}{{\hat{\overset{\sim}{C}}}_{{DOM},{CORR}}^{(d)}(k)}}} & {(31)} \\{:={\sum\limits_{d = 1}^{M}{{\Xi_{GRID}^{(d)}(k)}{\alpha^{(d)}(k)}{{\hat{\overset{\sim}{x}}}_{DOM}^{(d)}(k)}}}} & {(32)}\end{matrix}$

can be expressed in terms of the perceptual coding errors

{circumflex over ({tilde over (e)})}_(DOM) ^((d))(k):={tilde over(x)}_(DOM) ^((d))(k)−{circumflex over ({tilde over (x)})}_(DOM)^((d))(k)  (33)

in the individual directional signals by

{circumflex over ({tilde over (E)})}_(DIR) ^((M))(k)=Σ_(d=1)^(M)Ξ_(GRID) ^((d))(k)α^((d))(k){circumflex over ({tilde over(e)})}_(DOM) ^((d))(k).  (34)

The representation of the error {circumflex over ({tilde over(E)})}_(DIR) ^((M))(k) in the spatial domain with respect to the testdirections Ω_(q), q=1, . . . , Q, is given by

$\begin{matrix}{{{\hat{\overset{\sim}{W}}}_{{DIR},q}^{(M)}(d)} = {\sum\limits_{d = 1}^{M}\; {\underset{\underset{= {:{\beta^{(d)}{(k)}}}}{}}{\Xi^{T}{\Xi_{GRID}^{(d)}(k)}{\alpha^{(d)}(k)}}{{{\hat{\overset{\sim}{e}}}_{DOM}^{(d)}(k)}.}}}} & (35)\end{matrix}$

Denoting the elements of the vector β^((d))(k) by β_(q) ^((d))(k), q=1,. . . , Q, and assuming the individual perceptual coding errors{circumflex over ({tilde over (e)})}_(DOM) ^((d))(k), d=1, . . . , M, tobe independent of each other, it follows from equation (35) that theelements

_(DIR,q) ^((M))(k,b) of the directional power distribution

_(DIR) ^((M))(k,b) of the perceptual coding error {circumflex over({tilde over (E)})}_(DIR) ^((M))(k) can be computed by

_(DIR,q) ^((M))(k,b)=Σ_(d=1) ^(M)(β_(q) ^((d)) k))²{tilde over(σ)}_(DIR,d) ²(k,b)  (36)

{tilde over (σ)}_(DIR,d) ²(k,b) is supposed to represent the power ofthe perceptual quantisation error within the b-th critical band in thedirectional signal {circumflex over ({tilde over (x)})}_(DOM) ^((d))(k).This power can be assumed to correspond to the perceptual masking powerof the directional signal {tilde over (x)}_(DOM) ^((d))(k).

-   c. For computing the directional power distribution    _(AMB,RED) ^((M))(k,b) of the error {circumflex over ({tilde over    (E)})}_(AMB,RED) ^((M))(k) resulting from the perceptual coding of    the HOA coefficient sequences of the ambient HOA component, each HOA    coefficient sequence is assumed to be coded independently. Hence,    the errors introduced into the individual HOA coefficient sequences    within each Bark scale critical band can be assumed to be    uncorrelated. This means that the inter-coefficient correlation    matrix of the error {circumflex over ({tilde over (E)})}_(AMB,RED)    ^((M))(k) with respect to each Bark scale critical band is diagonal,    i.e.

{tilde over (Σ)}_(AMB,RED) ^((M))(k,b)=diag({tilde over (σ)}_(AMB,RED,1)^(2(M))(k,b),{tilde over (σ)}_(AMB,RED,2) ^(2(M))(k,b), . . . ,{tildeover (σ)}_(AMB,RED,O) ^(2(M))(k,b)).  (37)

The elements {tilde over (σ)}_(AMB,RED,o) ^(2(M))(k,b), o=1, . . . , O,are supposed to represent the power of the perceptual quantisation errorwithin the b-th critical band in the o-th coded HOA coefficient sequencein {circumflex over ({tilde over (C)})}_(AMB,RED) ^((M))(k). They can beassumed to correspond to the perceptual masking power of the o-th HOAcoefficient sequence {tilde over (C)}_(AMB,RED) ^((M))(k). Thedirectional power distribution of the perceptual coding error{circumflex over ({tilde over (E)})}_(AMB,RED) ^((M))(k) is thuscomputed by

_(AMB,RED) ^((M))(k,b)=diag(μ^(T){tilde over (Σ)}_(AMB,RED)^((M))(k,b)Ξ).  (38)

B. Improved HOA Decompression

The corresponding HOA decompression processing is depicted in FIG. 3 andincludes the following steps or stages. In step or stage 31 a perceptualdecoding of the I signals contained in Y̆(k−2) is performed in order toobtain the I decoded signals in Ŷ(k−2).

In signal re-distributing step or stage 32, the perceptually decodedsignals in Ŷ(k−2) are re-distributed in order to recreate the frame{circumflex over (X)}_(DIR)(k−2) of directional signals and the frameĈ_(AMB,RED)(k−2) of the ambient HOA component. The information about howto re-distribute the signals is obtained by reproducing the assigningoperation performed for the HOA compression, using the index data sets

_(DIR,ACT)(k) and

_(AMB,ACT)(k−2) Since this is a recursive procedure (see section A), theadditionally transmitted assignment vector γ(k) can be used in order toallow for an initialisation of the re-distribution procedure, e.g. incase the transmission is breaking down.

In composition step or stage 33, a current frame Ĉ(k−3) of the desiredtotal HOA representation is re-composed (according to the processingdescribed in connection with FIG. 2b and FIG. 4 of EP 12306569.0 usingthe frame {circumflex over (X)}_(DIR)(k−2) of the directional signals,the set

_(DIR,ACT)(k) of the active directional signal indices together with theset

_(Ω,ACT)(k) of the corresponding directions, the parameters ζ(k−2) forpredicting portions of the HOA representation from the directionalsignals, and the frame Ĉ_(AMB,RED)(k−2) of HOA coefficient sequences ofthe reduced ambient HOA component. Ĉ_(AMB,RED)(k−2) corresponds tocomponent {circumflex over (D)}_(A)(k−2) in EP 12306569.0, and

_(Ω,ACT) (k) and

_(DIR,ACT)(k) correspond to A_({circumflex over (Ω)})(k) in EP12306569.0, wherein active directional signal indices are marked in thematrix elements of A_({circumflex over (Ω)})(k). I.e., directionalsignals with respect to uniformly distributed directions are predictedfrom the directional signals ({circumflex over (X)}_(DIR)(k−2)) usingthe received parameters (ζ(k−2)) for such prediction, and thereafter thecurrent decompressed frame (Ĉ(k−3)) is re-composed from the frame ofdirectional signals ({circumflex over (X)}_(DIR)(k−2)), the predictedportions and the reduced ambient HOA component (Ĉ_(AMB,RED)(k−2)).

C. Basics of Higher Order Ambisonics

Higher Order Ambisonics (HOA) is based on the description of a soundfield within a compact area of interest, which is assumed to be free ofsound sources. In that case the spatiotemporal behaviour of the soundpressure p(t,x) at time t and position x within the area of interest isphysically fully determined by the homogeneous wave equation. In thefollowing a spherical coordinate system as shown in FIG. 4 is assumed.In the used coordinate system, the x axis points to the frontalposition, the y axis points to the left, and the z axis points to thetop. A position in space x=(r,θ,ϕ)^(T) is represented by a radius r>0(i.e. the distance to the coordinate origin), an inclination angleθ∈[0,π] measured from the polar axis z and an azimuth angle ϕ∈[0,2π[measured counter-clockwise in the x-y plane from the x axis. Further,(⋅)^(T) denotes the transposition.

It can be shown (see E. G. Williams, “Fourier Acoustics”, volume 93 ofApplied Mathematical Sciences, Academic Press, 1999) that the Fouriertransform of the sound pressure with respect to time denoted by

_(t)(⋅), i.e.

P(ω,x)=

_(t)(p(t,x))=∫_(−∞) ^(∞) p(t,x)e ^(−iωt) dt,  (39)

with ω denoting the angular frequency and i indicating the imaginaryunit, can be expanded into a series of Spherical Harmonics according to

P(ω=kc _(s) ,r,θ,ϕ)=Σ_(n=0) ^(N)Σ_(m=−n) ^(n) A _(n) ^(m)(k)j _(n)(kr)S_(n) ^(m)(θ,ϕ).  (40)

In equation (40), c_(s) denotes the speed of sound and k denotes theangular wave number, which is related to the angular frequency ω by

$k = {\frac{\omega}{c_{s}}.}$

Further, j_(n)(⋅) denote the spherical Bessel functions of the firstkind and S_(n) ^(m)(θ,ϕ) denote the real valued Spherical Harmonics oforder n and degree m, which are defined in below section C.1. Theexpansion coefficients A_(n) ^(m)(k) are depending only on the angularwave number k. In the foregoing it has been implicitly assumed thatsound pressure is spatially band-limited. Thus, the series of SphericalHarmonics is truncated with respect to the order index n at an upperlimit N, which is called the order of the HOA representation.

If the sound field is represented by a superposition of an infinitenumber of harmonic plane waves of different angular frequencies ωarriving from all possible directions specified by the angle tuple(θ,ϕ), it can be shown (see B. Rafaely, “Plane-wave Decomposition of theSound Field on a Sphere by Spherical Convolution”, Journal of theAcoustical Society of America, vol. 4 (116), pages 2149-2157, 2004) thatthe respective plane wave complex amplitude function C(ω,θ,ϕ) can beexpressed by the following Spherical Harmonics expansion

C(ω=kc _(s),θ,ϕ)=Σ_(n=0) ^(N)Σ_(m=−n) ^(n) C _(n) ^(m)(k)S _(n)^(m)(θ,ϕ),  (41)

where the expansion coefficients C_(n) ^(m)(k) are related to theexpansion coefficients

A _(n) ^(m)(k) by A _(n) ^(m)(k)=4πi ^(n) C _(n) ^(m)(k).   (42)

Assuming the individual coefficients C_(n) ^(m)(ω=kc_(s)) to befunctions of the angular frequency ω, the application of the inverseFourier transform (denoted by

⁻¹(⋅)) provides time domain functions

$\begin{matrix}{{c_{n}^{m}(t)} = {{\mathcal{F}_{t}^{- 1}\left( {C_{n}^{m}\left( {\omega \text{/}c_{s}} \right)} \right)} = {\frac{1}{2\pi}{\int_{- \infty}^{\infty}{{C_{n}^{m}\left( \frac{\omega}{c_{s}} \right)}e^{i\; \omega \; t}d\; \omega}}}}} & (43)\end{matrix}$

for each order n and degree m, which can be collected in a single vectorc(t) by

c(t)=[c ₀ ⁰(t) c ₁ ⁻¹(t) c ₁ ⁰(t) c ₁ ¹(t) c ₂ ⁻²(t) c ₂ ⁻¹(t) c ₂ ⁰(t)c ₂ ¹(t) c ₂ ²(t) . . . c _(N) ^(N−1)(t) c _(N) ^(N)(t)]^(T).  (44)

The position index of a time domain function c_(n) ^(m)(t) within thevector c(t) is given by n(n+1)+1+m. The overall number of elements invector c(t) is given by O=(N+1)².

The final Ambisonics format provides the sampled version of c(t) using asampling frequency f_(S) as

{

={c(T _(S)),c(2T _(S)),c(3T _(S)),c(4T _(S)), . . . }  (45)

where T_(S)=1/f_(S) denotes the sampling period. The elements ofc(lT_(S)) are here referred to as Ambisonics coefficients. The timedomain signals c_(n) ^(m)(t) and hence the Ambisonics coefficients arereal-valued.

C.1 Definition of Real-Valued Spherical Harmonics

The real-valued spherical harmonics S_(n) ^(m)(θ,ϕ) are given by

$\begin{matrix}{{S_{n}^{m}\left( {\theta,\varphi} \right)} = {\sqrt{\frac{\left( {{2\; n} + 1} \right)}{4\pi}\frac{\left( {n - {m}} \right)!}{\left( {n + {m}} \right)!}}{P_{n,{m}}\left( {\cos \; \theta} \right)}{{trg}_{m}(\varphi)}}} & (46) \\{{with}\text{}{{{trg}_{m}(\varphi)} = \left\{ {\begin{matrix}{\sqrt{2}{\cos \left( {m\; \varphi} \right)}} & {m > 0} \\1 & {m = 0} \\{{- \sqrt{2}}{\sin \left( {m\; \varphi} \right)}} & {m < 0}\end{matrix}.} \right.}} & (47)\end{matrix}$

The associated Legendre functions P_(n,m)(x) are defined as

$\begin{matrix}{{{P_{n,m}(x)} = {\left( {1 - x^{2}} \right)^{\frac{m}{2}}\frac{d^{m}}{{dx}^{m}}{P_{n}(x)}}},{m \geq 0}} & (48)\end{matrix}$

with the Legendre polynomial P_(n)(x) and, unlike in the above-mentionedWilliams article, without the Condon-Shortley phase term (−1)^(m).

C.2 Spatial Resolution of Higher Order Ambisonics

A general plane wave function x(t) arriving from a directionΩ₀=(θ₀,ϕ₀)^(T) is represented in HOA by

c _(n) ^(m)(t)=x(t)S _(n) ^(m)(Ω₀), 0≤n≤N, |m|≤n.  (49)

The corresponding spatial density of plane wave amplitudes c(t,Ω):=

_(t) ⁻¹(C(ω,Ω)) is given by

$\begin{matrix}{{c\left( {t,\Omega} \right)} = {\sum\limits_{n = 0}^{N}\; {\sum\limits_{m = {- n}}^{n}\; {{c_{n}^{m}(t)}{S_{n}^{m}(\Omega)}}}}} & {(50)} \\{= {{x(t)}{\underset{\underset{v_{N}{(\Theta)}}{}}{\left\lbrack {\sum\limits_{n = 0}^{N}\; {\sum\limits_{m = {- n}}^{n}{{S_{n}^{m}\left( \Omega_{0} \right)}{S_{n}^{m}(\Omega)}}}} \right\rbrack}.}}} & {(51)}\end{matrix}$

It can be seen from equation (51) that it is a product of the generalplane wave function x(t) and of a spatial dispersion function v_(N)(Θ),which can be shown to only depend on the angle Θ between Ω and Ω₀ havingthe property

cos Θ=cos θ cos θ₀+cos(ϕ−ϕ₀)sin θ sin θ₀.  (52)

As expected, in the limit of an infinite order, i.e., N→∞, the spatialdispersion function turns into a Dirac delta δ(⋅), i.e.

$\begin{matrix}{{\lim\limits_{N\rightarrow\infty}{v_{N}(\Theta)}} = {\frac{\delta (\Theta)}{2\; \pi}.}} & (53)\end{matrix}$

However, in the case of a finite order N, the contribution of thegeneral plane wave from direction Ω₀ is smeared to neighbouringdirections, where the extent of the blurring decreases with anincreasing order. A plot of the normalised function v_(N)(Θ) fordifferent values of N is shown in FIG. 5.

It should be pointed out that for any direction Ω the time domainbehaviour of the spatial density of plane wave amplitudes is a multipleof its behaviour at any other direction. In particular, the functionsc(t,Ω₁) and c(t,Ω₂) for some fixed directions Ω₁ and Ω₂ are highlycorrelated with each other with respect to time t.

C.3 Spherical Harmonic Transform

If the spatial density of plane wave amplitudes is discretised at anumber of O spatial directions Ω_(o), 1≤o≤0, which are nearly uniformlydistributed on the unit sphere, O directional signals c(t,Ω_(o)) areobtained. Collecting these signals into a vector as

c _(SPAT)(t):=[c(t,Ω ₁) . . . c(t,Ω _(o))]^(T),  (54)

by using equation (50) it can be verified that this vector can becomputed from the continuous Ambisonics representation d(t) defined inequation (44) by a simple matrix multiplication as

c _(SPAT)(t)=Ψ^(H) c(t),  (55)

where (⋅)^(H) indicates the joint transposition and conjugation, and Ψdenotes a mode-matrix defined by

Ψ:=[S ₁ . . . S ₀]  (56)

with

S _(o):=[S ₀ ⁰(Ω_(o)) S ₁ ⁻¹(Ω_(o)) S ₁ ⁰(Ω_(o)) S ₁ ¹(Ω_(o)) . . . S_(n) ^(N−1)(Ω_(o)) S _(N) ^(N)(Ω_(o))].  (57)

Because the directions Ω_(o) are nearly uniformly distributed on theunit sphere, the mode matrix is invertible in general. Hence, thecontinuous Ambisonics representation can be computed from thedirectional signals c(t,Ω_(o)) by

c(t)=Ψ^(−H) c _(SPAT)(t).  (58)

Both equations constitute a transform and an inverse transform betweenthe Ambisonics representation and the spatial domain. These transformsare here called the Spherical Harmonic Transform and the inverseSpherical Harmonic Transform.

It should be noted that since the directions Ω_(o) are nearly uniformlydistributed on the unit sphere, the approximation

Ψ^(H)≈Ψ⁻¹  (59)

is available, which justifies the use of Ψ⁻¹ instead of Ψ^(H) inequation (55).

Advantageously, all the mentioned relations are valid for thediscrete-time domain, too.

The inventive processing can be carried out by a single processor orelectronic circuit, or by several processors or electronic circuitsoperating in parallel and/or operating on different parts of theinventive processing.

1. A method for decompressing a compressed Higher Order Ambisonicsrepresentation, the method comprising: decoding a current encodedcompressed frame to provide a decoded frame of channels; re-distributingthe decoded frame of channels based on an assignment vector indicatingat least an index of a possibly contained coefficient sequence of anambient HOA component, and at least an index of active directionalsignals, wherein the re-distribution creates a frame of directionalsignals and a frame of an ambient HOA component; re-composing a currentdecompressed frame of the HOA representation from the frame ofdirectional signals and from the frame of the ambient HOA component; andwherein predicted signals with respect to uniformly distributeddirections are predicted from the directional signals, and the currentdecompressed frame is re-composed from the frame of the directionalsignals, the predicted signals and the frame of the ambient HOAcomponent.
 2. An apparatus for decompressing a Higher Order Ambisonicsrepresentation, the apparatus comprising: a processor for decoding acurrent encoded compressed frame to provide a decoded frame of channels;wherein the processor is further configured to re-distribute the decodedframe of channels based on an assignment vector indicating at least anindex of a possibly contained coefficient sequence of an ambient HOAcomponent, and at least an index of active directional signals, whereinthe re-distribution creates a frame of directional signals and a frameof an ambient HOA component; wherein the processor is further configuredto re-compose a current decompressed frame of the HOA representationfrom the frame of directional signals and from the frame of the ambientHOA component; and wherein predicted signals with respect to uniformlydistributed directions are predicted from the directional signals, andthe current decompressed frame is re-composed from the frame of thedirectional signals, the predicted signals and the frame of the ambientHOA component.
 3. A non-transitory computer readable storage mediumcontaining instructions that when executed by a processor perform amethod according to claim 1.